What I mean is, in the Hierarchy of Sins against Humanity there are entries for "manslaughter", for "murder". However, one might argue that the Kindred are already dead. So, would you argue that causing the Final Death of another Cainite constitute murder? What if the victim is a wight (i.e., a vampire with a Humanity/Path rating of 0, therefore arguably a mindless beast)?
I would say yes, killing another vampire calls for a degeneration roll and may lead to the loss of Humanity.
Vampires (on the path of Humanity, which we're concerned with here) consider themselves, or at least try to consider themselves (at least partially) human - that is what's reflected by their Humanity score. If a vampire strictly and unquestionably believed that vampires are completely dead as humans, said vampire would have to apply the very same thought to herself too, ie. consider herself dead as a human as well. Were she able to do so, it would mean she would have no Humanity left in her either at all (Humanity: 0.) (Once again, we're dealing with vampires on the Path of Humanity only here.)
As for destroying a vampire with a Humanity/Path rating of 0: I think vampire characters have no means of learning the Humanity score noted on the character sheet of a PC or an NPC. They might very well consider said vampire to be a madman, an utter degenerate worse than the worst of humanity... yet they would consider the vampire in question to be a mad, utterly degenerate... human+Beast combination (see the previous paragraph.) And killing a human, even if that human is a walking horror, may lead to the loss of your own Humanity (as you're getting closer to accepting and utilizing the horror's methods... even if to eliminate the horror.)
If the would-be destroyers somehow gained certain, unquestionable, in-game proof of some other vampire's zero Humanity rating and they gained similarly rock solid proof about the impossibility of vampiric redemption (your ST discarded Golconda?), they could probably kill the zero Humanity vampire without risking losing Humanity for the act. But I highly doubt such proof would be available in the stories of most Storytellers.
AFAIK that question is never answered on any corebooks. In V:tM, and other WoD many things are left to the discretion or imagination of the Storyteller, specially when it comes to ethic. So, this question can not be given an absolute answer, as there is no absolute ethic answer to that dilemma. It depends on the worldview of the Storyteller.
What would make killing a vampire do to the character mind?
Could the character feel remorse for killing a sentient being with feelings, memories, and its own motivations?
Would an innocent mind be affected after that kill?
If the answer is yes (I think that way), the character must make a degeneration roll to see how much innocent he preserves.
Notice that Humanity is not for punishing the character to do the wrong thing (at leas it's not always that). It's for roleplaying the character evolution as his values and ethics erodes. Sometimes doing the right thing will cost you Humanity.
If a character kills his best friend because he has become a wight, or because the Sabbat has abducted him and now he is a serial killer, probably killing him is the right thing to do, but I would make his player roll for degeneration.
I think the answer is actually a bit more complex than this. Does final deathing a vampire save others? Is the target vampire killing a lot of humans or causing significant human misery? Id adjudge that more as impassioned violation than murder.
Consider the consequences of non-intervention. If the enemy is incredibly powerful, there's little to blame a PC for not acting immediately (survival, risk of breaking the masquarade, etc).
Consider it as essentially the same thing as killing a mortal--vampires on this path are trying to hang on to their humanity and psychologically would be affected much the same way anytime they killed a thinking, feeling creature--human, vampire, or otherwise.
But killing isn't necessarily murder, and murder may be impulsive, planned, or even enjoyed by the perpetrator. A lot depends on that. For example, killing mortals. Killing a five-year-old child, versus killing a bum you were feeding on, versus assassinating Hitler... they're different things. It's got to be taken on a case-by-case basis. A vampire with a humanity of 7 probably wouldn't feel guilty for having to destroy a vampire that's lost their humanity altogether and gone on a killing spree, because that was the only way to save the lives of those potential victims. But they wouldn't enjoy it, either. It would be like shooting a rabid dog--a tragedy, not a victory.
At very high humanity, just killing at all--even the equivalent of killing Hitler or a 0-Humanity vampire--will cause regret. But 9 and 10 Humanity are supposed to be saintly, the sort of person who values life and wants to protect others; killing, even killing an absolute monster, goes against everything they believe in. These are the guys who would want to immobilize the 0-Humanity vampire and look for a way to cure them (and let's face it, at 0 humanity, it's not like they're going to be coming out of torpor anytime soon, so your saintly character just has to stash them away and go off to research the problem). I've never tried playing a character with a humanity score that high, but it seems like an interesting logistical challenge, always having to out-think not just your opponents but your own Beast.
Depending on the vampire, much of the time you probably won't have to worry about humanity loss because your character will be killing, for example, Sabbat vampires who are quite happy to feed in serial-killer fashion and will continue doing so if not stopped. If you're playing at a typical 6-7 Humanity, that's killing in defense of self or others and thus not murder; it doesn't degrade your humanity.
How cruel a ST do you want to be. And how "Humane" is the killer. If he had a humanity of say 10 and it was me he would have to make a degeneration check. However I know some ST where it would be perfectly fine. Also If you don't want him to lose humanity or you do and he passes the check then you can always have the Prince call a Blood Hunt for the killer. Thus solving the humanity issue. Unless of course the PC is more powerful than the Prince. Not likely but possible.